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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsmssspx 16101 Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.)

Theoremlsmpr 16102 The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)

Theoremlsppreli 16103 A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlsmelpr 16104 Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)

Theoremlsppr0 16105 The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)

Theoremlsppr 16106* Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
Scalar

Theoremlspprel 16107* Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
Scalar

Theoremlspprabs 16108 Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)

Theoremlspvadd 16109 The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntri 16110 Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntrim 16111 Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlbspropd 16112* If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Scalar       Scalar                                          LBasis LBasis

Theorempj1lmhm 16113 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
s LMHom

Theorempj1lmhm2 16114 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
s LMHom s

10.7  Vector spaces

10.7.1  Definition and basic properties

Syntaxclvec 16115 Extend class notation with class of all left vector spaces.

Definitiondf-lvec 16116 Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremislvec 16117 The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremlvecdrng 16118 The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
Scalar

Theoremlveclmod 16119 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)

Theoremlsslvec 16120 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
s

Theoremlvecvs0or 16121 If a scalar product is zero, one of its factors must be zero. (hvmul0or 22389 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvsn0 16122 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
Scalar

Theoremlssvs0or 16123 If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs. (Contributed by NM, 5-Apr-2015.)
Scalar

Theoremlvecvscan 16124 Cancellation law for scalar multiplication. (hvmulcan 22436 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvscan2 16125 Cancellation law for scalar multiplication. (hvmulcan2 22437 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecinv 16126 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
Scalar

Theoremlspsnvs 16127 A non-zero scalar product does not change the span of a singleton. (spansncol 22932 analog.) (Contributed by NM, 23-Apr-2014.)
Scalar

Theoremlspsneleq 16128 Membership relation that implies equality of spans. (spansneleq 22934 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspsncmp 16129 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)

Theoremlspsnne1 16130 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)

Theoremlspsnne2 16131 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)

Theoremlspsnnecom 16132 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)

Theoremlspabs2 16133 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspabs3 16134 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspsneq 16135* Equal spans of singletons must have proportional vectors. See lspsnss2 16022 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
Scalar

Theoremlspsneu 16136* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
Scalar

Theoremlspsnel4 16137 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 22937 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspdisj 16138 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)

Theoremlspdisjb 16139 A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)

Theoremlspdisj2 16140 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)

Theoremlspfixed 16141* Show membership in the span of the sum of two vectors, one of which () is fixed in advance. (Contributed by NM, 27-May-2015.)

Theoremlspexch 16142 Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 16143 vs. lspexchn2 16144); look for lspexch 16142 and prcom 3839 in same proof. TODO: would a hypothesis of instead of { Z } ) ` be better overall? This would be shorter and also satisfy the condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)

Theoremlspexchn1 16143 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 16142 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)

Theoremlspexchn2 16144 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 16142 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)

Theoremlspindpi 16145 Partial independence property. (Contributed by NM, 23-Apr-2015.)

Theoremlspindp1 16146 Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)

Theoremlspindp2l 16147 Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)

Theoremlspindp2 16148 Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)

Theoremlspindp3 16149 Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlspindp4 16150 (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlvecindp 16151 Compute the coefficient in a sum with an independent vector (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions and (second conjunct). Typically, is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Scalar

Theoremlvecindp2 16152 Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
Scalar

Theoremlspsnsubn0 16153 Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)

Theoremlsmcv 16154 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 23016 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)

Theoremlspsolvlem 16155* Lemma for lspsolv 16156. (Contributed by Mario Carneiro, 25-Jun-2014.)
Scalar

Theoremlspsolv 16156 If is in the span of but not , then is in the span of . (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremlssacsex 16157* In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 15984 by lspsolv 16156. (Contributed by David Moews, 1-May-2017.)
mrCls              ACS

Theoremlspsnat 16158 There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 22945 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)

Theoremlspsncv0 16159* The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.)

Theoremlsppratlem1 16160 Lemma for lspprat 16166. Let (if there is no such then is the zero subspace), and let (assuming the conclusion is false). The goal is to write , in terms of , , which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 16156 (hence the name), which we use extensively below. In this lemma, we show that since , either or . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem2 16161 Lemma for lspprat 16166. Show that if and are both in (which will be our goal for each of the two cases above), then , contradicting the hypothesis for . (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.)

Theoremlsppratlem3 16162 Lemma for lspprat 16166. In the first case of lsppratlem1 16160, since , also , and since and , we have as desired. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem4 16163 Lemma for lspprat 16166. In the second case of lsppratlem1 16160, and implies and thus as well. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem5 16164 Lemma for lspprat 16166. Combine the two cases and show a contradiction to under the assumptions on and . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem6 16165 Lemma for lspprat 16166. Negating the assumption on , we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)

Theoremlspprat 16166* A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)

Theoremislbs2 16167* An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
LBasis

Theoremislbs3 16168* An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlbsacsbs 16169 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 16167. (Contributed by David Moews, 1-May-2017.)
mrCls              mrInd       LBasis

Theoremlvecdim 16170 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 16157 and lbsacsbs 16169 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 14550. (Contributed by David Moews, 1-May-2017.)
LBasis

Theoremlbsextlem1 16171* Lemma for lbsext 16176. The set is the set of all linearly independent sets containing ; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlbsextlem2 16172* Lemma for lbsext 16176. Since is a chain (actually, we only need it to be closed under binary union), the union of the spans of each individual element of is a subspace, and it contains all of (except for our target vector - we are trying to make a linear combination of all the other vectors in some set from ). (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis                                                               []

Theoremlbsextlem3 16173* Lemma for lbsext 16176. A chain in has an upper bound in . (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis                                                               []

Theoremlbsextlem4 16174* Lemma for lbsext 16176. lbsextlem3 16173 satisfies the conditions for the application of Zorn's lemma zorn 8334 (thus invoking AC), and so there is a maximal linearly independent set extending . Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlbsextg 16175* For any linearly independent subset of , there is a basis containing the vectors in . (Contributed by Mario Carneiro, 17-May-2015.)
LBasis

Theoremlbsext 16176* For any linearly independent subset of , there is a basis containing the vectors in . (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
LBasis

Theoremlbsexg 16177 Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
LBasis       CHOICE

Theoremlbsex 16178 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlvecprop2d 16179* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 16180 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlvecpropd 16180* If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

10.8  Ideals

10.8.1  The subring algebra; ideals

Syntaxcsra 16181 Extend class notation with the subring algebra generator.
subringAlg

Syntaxcrglmod 16182 Extend class notation with the left module induced by a ring over itself.
ringLMod

Syntaxclidl 16183 Ring left-ideal function.
LIdeal

Syntaxcrsp 16184 Ring span function.
RSpan

Definitiondf-sra 16185* Given any subring of a ring, we can construct a left-algebra by regarding the elements of the subring as scalars and the ring itself as a set of vectors. (Contributed by Mario Carneiro, 27-Nov-2014.)
subringAlg sSet Scalar s sSet

Definitiondf-rgmod 16186 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod subringAlg

Definitiondf-lidl 16187 Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
LIdeal ringLMod

Definitiondf-rsp 16188 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan ringLMod

Theoremsraval 16189 Lemma for srabase 16191 through sravsca 16195. (Contributed by Mario Carneiro, 27-Nov-2014.)
subringAlg sSet Scalar s sSet

Theoremsralem 16190 Lemma for srabase 16191 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               Slot

Theoremsrabase 16191 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsraaddg 16192 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsramulr 16193 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsrasca 16194 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg               s Scalar

Theoremsravsca 16195 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsratset 16196 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               TopSet TopSet

Theoremsratopn 16197 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsrads 16198 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsralmod 16199 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
subringAlg        SubRing

Theoremsralmod0 16200 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
subringAlg

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